Now the same arguments used to establish invariance of the classical
Yang-Mills action functional under the symmetry \(\varphi \)  show that
the symplectic form (REF ) is preserved by
\(\varphi \) . Thus the partition function (REF )
is formally invariant under the symmetry \(\varphi \) . However, the
definition of the formal path integral (REF )
requires a suitable regularisation to make it mathematically well-defined, and it may
be that there is no regularisation which preserves the symmetry; in
such a case the symmetry is anomalous and the partition function is not
invariant. We will show in Section REF  below, using the
lattice regularisation of two-dimensional Yang-Mills theory, that
indeed the quantum gauge theory is also invariant under the symmetry
induced by the outer automorphism \(\varphi :G\longrightarrow G\) . Different prescriptions for defining the path integral in
(REF ) will differ by a
renormalisation ambiguity depending on the topology and area of
\(\Sigma \)  as [1]}, [2]}
\(\Delta S =\upsilon _1\, \chi (\Sigma )+\upsilon _2\, e^2\, a\) 
